LMFDB - Embedded newform 3060.2.w.d.2177.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [3060,2,Mod(953,3060)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("3060.953"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3060, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([0, 2, 3, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 3060 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3060.w (of order $4$, degree $2$, minimal)

Newform invariants

comment:select newform

sage:traces = [8,0,0,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$24.4342230185$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(i)$
Coefficient field:	$\Q(\zeta_{24})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{4} + 1 $ x^8 - x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			2177.3
Root			$0.258819 - 0.965926i$ of defining polynomial
Character	$\chi$	$=$	3060.2177
Dual form			3060.2.w.d.953.4

$q$-expansion

comment:q-expansion

sage:f.q_expansion() # note that sage often uses an isomorphic number field

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.41421 - 1.73205i) q^{5} +(1.22474 + 1.22474i) q^{7} -3.14626i q^{11} +(-2.00000 + 2.00000i) q^{13} +(-0.707107 + 0.707107i) q^{17} +1.00000i q^{19} +(-5.97469 - 5.97469i) q^{23} +(-1.00000 - 4.89898i) q^{25} +5.97469 q^{29} +5.34847 q^{31} +(3.85337 - 0.389270i) q^{35} +(-3.22474 - 3.22474i) q^{37} -5.97469i q^{41} +(2.44949 - 2.44949i) q^{43} +(4.48905 - 4.48905i) q^{47} -4.00000i q^{49} +(-2.43916 - 2.43916i) q^{53} +(-5.44949 - 4.44949i) q^{55} +5.51399 q^{59} -3.34847 q^{61} +(0.635674 + 6.29253i) q^{65} +(-4.44949 - 4.44949i) q^{67} +0.635674i q^{71} +(-2.32577 + 2.32577i) q^{73} +(3.85337 - 3.85337i) q^{77} +7.55051i q^{79} +(6.92820 + 6.92820i) q^{83} +(0.224745 + 2.22474i) q^{85} -0.142865 q^{89} -4.89898 q^{91} +(1.73205 + 1.41421i) q^{95} +(-4.89898 - 4.89898i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 16 q^{13} - 8 q^{25} - 16 q^{31} - 16 q^{37} - 24 q^{55} + 32 q^{61} - 16 q^{67} - 48 q^{73} - 8 q^{85}+O(q^{100}) $ 8 * q - 16 * q^13 - 8 * q^25 - 16 * q^31 - 16 * q^37 - 24 * q^55 + 32 * q^61 - 16 * q^67 - 48 * q^73 - 8 * q^85

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3060\mathbb{Z}\right)^\times$.

$n$	$1261$	$1361$	$1531$	$1837$
$\chi(n)$	$1$	$-1$	$1$	$e\left(\frac{1}{4}\right)$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	1.41421	−	1.73205i	0.632456	−	0.774597i
$5$	1.41421	−	1.73205i	0.632456	−	0.774597i
$6$		0			0
$7$	1.22474	+	1.22474i	0.462910	+	0.462910i	0.899608	−	0.436698i	$-0.143852\pi$
$7$	1.22474	+	1.22474i	0.462910	+	0.462910i	−0.436698	+	0.899608i	$0.643852\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$		−	3.14626i		−	0.948634i	−0.880354	−	0.474317i	$-0.842695\pi$
$11$		−	3.14626i		−	0.948634i	0.880354	−	0.474317i	$-0.157305\pi$
$12$		0			0
$13$	−2.00000	+	2.00000i	−0.554700	+	0.554700i	−0.927794	−	0.373094i	$-0.878297\pi$
$13$	−2.00000	+	2.00000i	−0.554700	+	0.554700i	0.373094	+	0.927794i	$0.378297\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−0.707107	+	0.707107i	−0.171499	+	0.171499i
$17$	−0.707107	+	0.707107i	−0.171499	+	0.171499i
$18$		0			0
$19$			1.00000i			0.229416i	0.993399	+	0.114708i	$0.0365932\pi$
$19$			1.00000i			0.229416i	−0.993399	+	0.114708i	$0.963407\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−5.97469	−	5.97469i	−1.24581	−	1.24581i	−0.957553	−	0.288256i	$-0.906925\pi$
$23$	−5.97469	−	5.97469i	−1.24581	−	1.24581i	−0.288256	−	0.957553i	$-0.593075\pi$
$24$		0			0
$25$	−1.00000	−	4.89898i	−0.200000	−	0.979796i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	5.97469			1.10947			0.554736	−	0.832026i	$-0.312819\pi$
$29$	5.97469			1.10947			0.554736	+	0.832026i	$0.312819\pi$
$30$		0			0
$31$	5.34847			0.960613			0.480307	−	0.877101i	$-0.340525\pi$
$31$	5.34847			0.960613			0.480307	+	0.877101i	$0.340525\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	3.85337	−	0.389270i	0.651339	−	0.0657986i
$36$		0			0
$37$	−3.22474	−	3.22474i	−0.530145	−	0.530145i	0.390471	−	0.920615i	$-0.372312\pi$
$37$	−3.22474	−	3.22474i	−0.530145	−	0.530145i	−0.920615	+	0.390471i	$0.872312\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		−	5.97469i		−	0.933090i	−0.884497	−	0.466545i	$-0.845499\pi$
$41$		−	5.97469i		−	0.933090i	0.884497	−	0.466545i	$-0.154501\pi$
$42$		0			0
$43$	2.44949	−	2.44949i	0.373544	−	0.373544i	−0.495222	−	0.868766i	$-0.664913\pi$
$43$	2.44949	−	2.44949i	0.373544	−	0.373544i	0.868766	+	0.495222i	$0.164913\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	4.48905	−	4.48905i	0.654795	−	0.654795i	−0.299349	−	0.954144i	$-0.596770\pi$
$47$	4.48905	−	4.48905i	0.654795	−	0.654795i	0.954144	+	0.299349i	$0.0967695\pi$
$48$		0			0
$49$		−	4.00000i		−	0.571429i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−2.43916	−	2.43916i	−0.335044	−	0.335044i	0.519454	−	0.854498i	$-0.326135\pi$
$53$	−2.43916	−	2.43916i	−0.335044	−	0.335044i	−0.854498	+	0.519454i	$0.826135\pi$
$54$		0			0
$55$	−5.44949	−	4.44949i	−0.734809	−	0.599969i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	5.51399			0.717860			0.358930	−	0.933364i	$-0.383142\pi$
$59$	5.51399			0.717860			0.358930	+	0.933364i	$0.383142\pi$
$60$		0			0
$61$	−3.34847			−0.428728			−0.214364	−	0.976754i	$-0.568768\pi$
$61$	−3.34847			−0.428728			−0.214364	+	0.976754i	$0.568768\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	0.635674	+	6.29253i	0.0788457	+	0.780492i
$66$		0			0
$67$	−4.44949	−	4.44949i	−0.543592	−	0.543592i	0.380988	−	0.924580i	$-0.375584\pi$
$67$	−4.44949	−	4.44949i	−0.543592	−	0.543592i	−0.924580	+	0.380988i	$0.875584\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			0.635674i			0.0754407i	0.999288	+	0.0377203i	$0.0120096\pi$
$71$			0.635674i			0.0754407i	−0.999288	+	0.0377203i	$0.987990\pi$
$72$		0			0
$73$	−2.32577	+	2.32577i	−0.272210	+	0.272210i	−0.829989	−	0.557779i	$-0.811654\pi$
$73$	−2.32577	+	2.32577i	−0.272210	+	0.272210i	0.557779	+	0.829989i	$0.311654\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	3.85337	−	3.85337i	0.439132	−	0.439132i
$78$		0			0
$79$			7.55051i			0.849499i	0.905311	+	0.424749i	$0.139638\pi$
$79$			7.55051i			0.849499i	−0.905311	+	0.424749i	$0.860362\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	6.92820	+	6.92820i	0.760469	+	0.760469i	0.976407	−	0.215938i	$-0.0692809\pi$
$83$	6.92820	+	6.92820i	0.760469	+	0.760469i	−0.215938	+	0.976407i	$0.569281\pi$
$84$		0			0
$85$	0.224745	+	2.22474i	0.0243770	+	0.241307i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−0.142865			−0.0151436			−0.00757181	−	0.999971i	$-0.502410\pi$
$89$	−0.142865			−0.0151436			−0.00757181	+	0.999971i	$0.502410\pi$
$90$		0			0
$91$	−4.89898			−0.513553
$92$		0			0
$93$		0			0
$94$		0			0
$95$	1.73205	+	1.41421i	0.177705	+	0.145095i
$96$		0			0
$97$	−4.89898	−	4.89898i	−0.497416	−	0.497416i	0.413217	−	0.910633i	$-0.364405\pi$
$97$	−4.89898	−	4.89898i	−0.497416	−	0.497416i	−0.910633	+	0.413217i	$0.864405\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$		−	1.55708i		−	0.154935i	−0.996995	−	0.0774675i	$-0.975317\pi$
$101$		−	1.55708i		−	0.154935i	0.996995	−	0.0774675i	$-0.0246834\pi$
$102$		0			0
$103$	5.44949	−	5.44949i	0.536954	−	0.536954i	−0.385679	−	0.922633i	$-0.626033\pi$
$103$	5.44949	−	5.44949i	0.536954	−	0.536954i	0.922633	+	0.385679i	$0.126033\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	6.61037	−	6.61037i	0.639048	−	0.639048i	−0.311272	−	0.950321i	$-0.600755\pi$
$107$	6.61037	−	6.61037i	0.639048	−	0.639048i	0.950321	+	0.311272i	$0.100755\pi$
$108$		0			0
$109$			3.10102i			0.297024i	0.988911	+	0.148512i	$0.0474483\pi$
$109$			3.10102i			0.297024i	−0.988911	+	0.148512i	$0.952552\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−6.61037	−	6.61037i	−0.621851	−	0.621851i	0.324154	−	0.946004i	$-0.394921\pi$
$113$	−6.61037	−	6.61037i	−0.621851	−	0.621851i	−0.946004	+	0.324154i	$0.894921\pi$
$114$		0			0
$115$	−18.7980	+	1.89898i	−1.75292	+	0.177081i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−1.73205			−0.158777
$120$		0			0
$121$	1.10102			0.100093
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−9.89949	−	5.19615i	−0.885438	−	0.464758i
$126$		0			0
$127$	−4.10102	−	4.10102i	−0.363907	−	0.363907i	0.501342	−	0.865249i	$-0.332840\pi$
$127$	−4.10102	−	4.10102i	−0.363907	−	0.363907i	−0.865249	+	0.501342i	$0.832840\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$			7.56388i			0.660859i	0.943831	+	0.330430i	$0.107194\pi$
$131$			7.56388i			0.660859i	−0.943831	+	0.330430i	$0.892806\pi$
$132$		0			0
$133$	−1.22474	+	1.22474i	−0.106199	+	0.106199i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	6.53893	−	6.53893i	0.558659	−	0.558659i	−0.370267	−	0.928925i	$-0.620734\pi$
$137$	6.53893	−	6.53893i	0.558659	−	0.558659i	0.928925	+	0.370267i	$0.120734\pi$
$138$		0			0
$139$		−	2.44949i		−	0.207763i	−0.994590	−	0.103882i	$-0.966874\pi$
$139$		−	2.44949i		−	0.207763i	0.994590	−	0.103882i	$-0.0331263\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	6.29253	+	6.29253i	0.526208	+	0.526208i
$144$		0			0
$145$	8.44949	−	10.3485i	0.701692	−	0.859394i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−1.41421			−0.115857			−0.0579284	−	0.998321i	$-0.518450\pi$
$149$	−1.41421			−0.115857			−0.0579284	+	0.998321i	$0.518450\pi$
$150$		0			0
$151$	−1.89898			−0.154537			−0.0772684	−	0.997010i	$-0.524620\pi$
$151$	−1.89898			−0.154537			−0.0772684	+	0.997010i	$0.524620\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	7.56388	−	9.26382i	0.607545	−	0.744088i
$156$		0			0
$157$	−11.8990	−	11.8990i	−0.949642	−	0.949642i	0.0491495	−	0.998791i	$-0.484349\pi$
$157$	−11.8990	−	11.8990i	−0.949642	−	0.949642i	−0.998791	+	0.0491495i	$0.984349\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		−	14.6349i		−	1.15340i
$162$		0			0
$163$	−4.32577	+	4.32577i	−0.338820	+	0.338820i	−0.855923	−	0.517103i	$-0.827010\pi$
$163$	−4.32577	+	4.32577i	−0.338820	+	0.338820i	0.517103	+	0.855923i	$0.327010\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−4.73545	+	4.73545i	−0.366440	+	0.366440i	−0.866177	−	0.499737i	$-0.833430\pi$
$167$	−4.73545	+	4.73545i	−0.366440	+	0.366440i	0.499737	+	0.866177i	$0.333430\pi$
$168$		0			0
$169$			5.00000i			0.384615i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	4.87832	+	4.87832i	0.370891	+	0.370891i	0.867802	−	0.496910i	$-0.165532\pi$
$173$	4.87832	+	4.87832i	0.370891	+	0.370891i	−0.496910	+	0.867802i	$0.665532\pi$
$174$		0			0
$175$	4.77526	−	7.22474i	0.360975	−	0.546139i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	22.3417			1.66990			0.834948	−	0.550329i	$-0.185498\pi$
$179$	22.3417			1.66990			0.834948	+	0.550329i	$0.185498\pi$
$180$		0			0
$181$	−6.89898			−0.512797			−0.256399	−	0.966571i	$-0.582536\pi$
$181$	−6.89898			−0.512797			−0.256399	+	0.966571i	$0.582536\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−10.1459	+	1.02494i	−0.745941	+	0.0753554i
$186$		0			0
$187$	2.22474	+	2.22474i	0.162689	+	0.162689i
$188$		0			0
$189$		0			0
$190$		0			0
$191$		−	11.8065i		−	0.854290i	−0.904183	−	0.427145i	$-0.859519\pi$
$191$		−	11.8065i		−	0.854290i	0.904183	−	0.427145i	$-0.140481\pi$
$192$		0			0
$193$	−3.55051	+	3.55051i	−0.255571	+	0.255571i	−0.823250	−	0.567679i	$-0.807842\pi$
$193$	−3.55051	+	3.55051i	−0.255571	+	0.255571i	0.567679	+	0.823250i	$0.307842\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	19.1954	−	19.1954i	1.36762	−	1.36762i	0.503793	−	0.863825i	$-0.331938\pi$
$197$	19.1954	−	19.1954i	1.36762	−	1.36762i	0.863825	−	0.503793i	$-0.168062\pi$
$198$		0			0
$199$			24.6969i			1.75072i	0.483472	+	0.875360i	$0.339375\pi$
$199$			24.6969i			1.75072i	−0.483472	+	0.875360i	$0.660625\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	7.31747	+	7.31747i	0.513586	+	0.513586i
$204$		0			0
$205$	−10.3485	−	8.44949i	−0.722768	−	0.590138i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	3.14626			0.217632
$210$		0			0
$211$	5.34847			0.368204			0.184102	−	0.982907i	$-0.441062\pi$
$211$	5.34847			0.368204			0.184102	+	0.982907i	$0.441062\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−0.778539	−	7.70674i	−0.0530959	−	0.525595i
$216$		0			0
$217$	6.55051	+	6.55051i	0.444678	+	0.444678i
$218$		0			0
$219$		0			0
$220$		0			0
$221$		−	2.82843i		−	0.190261i
$222$		0			0
$223$	−5.44949	+	5.44949i	−0.364925	+	0.364925i	−0.865622	−	0.500698i	$-0.833077\pi$
$223$	−5.44949	+	5.44949i	−0.364925	+	0.364925i	0.500698	+	0.865622i	$0.333077\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−4.87832	+	4.87832i	−0.323785	+	0.323785i	−0.850217	−	0.526432i	$-0.823529\pi$
$227$	−4.87832	+	4.87832i	−0.323785	+	0.323785i	0.526432	+	0.850217i	$0.323529\pi$
$228$		0			0
$229$		−	25.6969i		−	1.69810i	−0.528311	−	0.849051i	$-0.677175\pi$
$229$		−	25.6969i		−	1.69810i	0.528311	−	0.849051i	$-0.322825\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	3.32124	+	3.32124i	0.217581	+	0.217581i	0.807478	−	0.589897i	$-0.200832\pi$
$233$	3.32124	+	3.32124i	0.217581	+	0.217581i	−0.589897	+	0.807478i	$0.700832\pi$
$234$		0			0
$235$	−1.42679	−	14.1237i	−0.0930732	−	0.921330i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	12.5851			0.814060			0.407030	−	0.913415i	$-0.366564\pi$
$239$	12.5851			0.814060			0.407030	+	0.913415i	$0.366564\pi$
$240$		0			0
$241$	−10.8990			−0.702065			−0.351032	−	0.936363i	$-0.614169\pi$
$241$	−10.8990			−0.702065			−0.351032	+	0.936363i	$0.614169\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−6.92820	−	5.65685i	−0.442627	−	0.361403i
$246$		0			0
$247$	−2.00000	−	2.00000i	−0.127257	−	0.127257i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		−	7.21393i		−	0.455339i	−0.973738	−	0.227670i	$-0.926889\pi$
$251$		−	7.21393i		−	0.455339i	0.973738	−	0.227670i	$-0.0731106\pi$
$252$		0			0
$253$	−18.7980	+	18.7980i	−1.18182	+	1.18182i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−11.1708	+	11.1708i	−0.696818	+	0.696818i	−0.963723	−	0.266905i	$-0.913999\pi$
$257$	−11.1708	+	11.1708i	−0.696818	+	0.696818i	0.266905	+	0.963723i	$0.413999\pi$
$258$		0			0
$259$		−	7.89898i		−	0.490819i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	3.99624	+	3.99624i	0.246418	+	0.246418i	0.819499	−	0.573081i	$-0.194252\pi$
$263$	3.99624	+	3.99624i	0.246418	+	0.246418i	−0.573081	+	0.819499i	$0.694252\pi$
$264$		0			0
$265$	−7.67423	+	0.775255i	−0.471424	+	0.0476235i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−27.6807			−1.68772			−0.843861	−	0.536562i	$-0.819723\pi$
$269$	−27.6807			−1.68772			−0.843861	+	0.536562i	$0.819723\pi$
$270$		0			0
$271$	31.3939			1.90704			0.953521	−	0.301326i	$-0.0974293\pi$
$271$	31.3939			1.90704			0.953521	+	0.301326i	$0.0974293\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−15.4135	+	3.14626i	−0.929468	+	0.189727i
$276$		0			0
$277$	−11.3485	−	11.3485i	−0.681863	−	0.681863i	0.278556	−	0.960420i	$-0.410144\pi$
$277$	−11.3485	−	11.3485i	−0.681863	−	0.681863i	−0.960420	+	0.278556i	$0.910144\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		−	30.6841i		−	1.83046i	−0.402932	−	0.915230i	$-0.632009\pi$
$281$		−	30.6841i		−	1.83046i	0.402932	−	0.915230i	$-0.367991\pi$
$282$		0			0
$283$	5.67423	−	5.67423i	0.337298	−	0.337298i	−0.518051	−	0.855350i	$-0.673342\pi$
$283$	5.67423	−	5.67423i	0.337298	−	0.337298i	0.855350	+	0.518051i	$0.173342\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	7.31747	−	7.31747i	0.431937	−	0.431937i
$288$		0			0
$289$		−	1.00000i		−	0.0588235i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−12.0529	−	12.0529i	−0.704139	−	0.704139i	0.261157	−	0.965296i	$-0.415896\pi$
$293$	−12.0529	−	12.0529i	−0.704139	−	0.704139i	−0.965296	+	0.261157i	$0.915896\pi$
$294$		0			0
$295$	7.79796	−	9.55051i	0.454015	−	0.556052i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	23.8988			1.38210
$300$		0			0
$301$	6.00000			0.345834
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−4.73545	+	5.79972i	−0.271151	+	0.332091i
$306$		0			0
$307$	−13.7980	−	13.7980i	−0.787491	−	0.787491i	0.193591	−	0.981082i	$-0.437986\pi$
$307$	−13.7980	−	13.7980i	−0.787491	−	0.787491i	−0.981082	+	0.193591i	$0.937986\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		−	3.14626i		−	0.178408i	−0.996013	−	0.0892041i	$-0.971568\pi$
$311$		−	3.14626i		−	0.178408i	0.996013	−	0.0892041i	$-0.0284324\pi$
$312$		0			0
$313$	−4.77526	+	4.77526i	−0.269913	+	0.269913i	−0.829065	−	0.559152i	$-0.811127\pi$
$313$	−4.77526	+	4.77526i	−0.269913	+	0.269913i	0.559152	+	0.829065i	$0.311127\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−7.21393	+	7.21393i	−0.405175	+	0.405175i	−0.880052	−	0.474877i	$-0.842492\pi$
$317$	−7.21393	+	7.21393i	−0.405175	+	0.405175i	0.474877	+	0.880052i	$0.342492\pi$
$318$		0			0
$319$		−	18.7980i		−	1.05248i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−0.707107	−	0.707107i	−0.0393445	−	0.0393445i
$324$		0			0
$325$	11.7980	+	7.79796i	0.654433	+	0.432553i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	10.9959			0.606222
$330$		0			0
$331$	−0.797959			−0.0438598			−0.0219299	−	0.999760i	$-0.506981\pi$
$331$	−0.797959			−0.0438598			−0.0219299	+	0.999760i	$0.506981\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−13.9993	+	1.41421i	−0.764862	+	0.0772667i
$336$		0			0
$337$	22.3712	+	22.3712i	1.21864	+	1.21864i	0.968110	+	0.250525i	$0.0806034\pi$
$337$	22.3712	+	22.3712i	1.21864	+	1.21864i	0.250525	+	0.968110i	$0.419397\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		−	16.8277i		−	0.911271i
$342$		0			0
$343$	13.4722	−	13.4722i	0.727430	−	0.727430i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−20.9275	+	20.9275i	−1.12345	+	1.12345i	−0.132226	+	0.991220i	$0.542212\pi$
$347$	−20.9275	+	20.9275i	−1.12345	+	1.12345i	−0.991220	+	0.132226i	$0.957788\pi$
$348$		0			0
$349$			25.8990i			1.38634i	0.720774	+	0.693170i	$0.243787\pi$
$349$			25.8990i			1.38634i	−0.720774	+	0.693170i	$0.756213\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	16.4384	+	16.4384i	0.874929	+	0.874929i	0.993005	−	0.118075i	$-0.0376724\pi$
$353$	16.4384	+	16.4384i	0.874929	+	0.874929i	−0.118075	+	0.993005i	$0.537672\pi$
$354$		0			0
$355$	1.10102	+	0.898979i	0.0584361	+	0.0477129i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	29.5556			1.55989			0.779943	−	0.625851i	$-0.215248\pi$
$359$	29.5556			1.55989			0.779943	+	0.625851i	$0.215248\pi$
$360$		0			0
$361$	18.0000			0.947368
$362$		0			0
$363$		0			0
$364$		0			0
$365$	0.739215	+	7.31747i	0.0386923	+	0.383014i
$366$		0			0
$367$	22.0000	+	22.0000i	1.14839	+	1.14839i	0.986869	+	0.161521i	$0.0516401\pi$
$367$	22.0000	+	22.0000i	1.14839	+	1.14839i	0.161521	+	0.986869i	$0.448360\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		−	5.97469i		−	0.310191i
$372$		0			0
$373$	−12.5505	+	12.5505i	−0.649841	+	0.649841i	−0.952954	−	0.303114i	$-0.901974\pi$
$373$	−12.5505	+	12.5505i	−0.649841	+	0.649841i	0.303114	+	0.952954i	$0.401974\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−11.9494	+	11.9494i	−0.615425	+	0.615425i
$378$		0			0
$379$		−	16.2474i		−	0.834575i	−0.908774	−	0.417288i	$-0.862981\pi$
$379$		−	16.2474i		−	0.834575i	0.908774	−	0.417288i	$-0.137019\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	3.21770	+	3.21770i	0.164417	+	0.164417i	0.784520	−	0.620103i	$-0.212909\pi$
$383$	3.21770	+	3.21770i	0.164417	+	0.164417i	−0.620103	+	0.784520i	$0.712909\pi$
$384$		0			0
$385$	−1.22474	−	12.1237i	−0.0624188	−	0.617882i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−27.5057			−1.39460			−0.697298	−	0.716781i	$-0.745615\pi$
$389$	−27.5057			−1.39460			−0.697298	+	0.716781i	$0.745615\pi$
$390$		0			0
$391$	8.44949			0.427309
$392$		0			0
$393$		0			0
$394$		0			0
$395$	13.0779	+	10.6780i	0.658019	+	0.537270i
$396$		0			0
$397$	18.3712	+	18.3712i	0.922023	+	0.922023i	0.997172	−	0.0751495i	$-0.0239434\pi$
$397$	18.3712	+	18.3712i	0.922023	+	0.922023i	−0.0751495	+	0.997172i	$0.523943\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$		−	31.1448i		−	1.55530i	−0.628699	−	0.777649i	$-0.716412\pi$
$401$		−	31.1448i		−	1.55530i	0.628699	−	0.777649i	$-0.283588\pi$
$402$		0			0
$403$	−10.6969	+	10.6969i	−0.532852	+	0.532852i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−10.1459	+	10.1459i	−0.502914	+	0.502914i
$408$		0			0
$409$			33.6969i			1.66621i	0.553118	+	0.833103i	$0.313438\pi$
$409$			33.6969i			1.66621i	−0.553118	+	0.833103i	$0.686562\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	6.75323	+	6.75323i	0.332305	+	0.332305i
$414$		0			0
$415$	21.7980	−	2.20204i	1.07002	−	0.108094i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	16.0171			0.782484			0.391242	−	0.920288i	$-0.372046\pi$
$419$	16.0171			0.782484			0.391242	+	0.920288i	$0.372046\pi$
$420$		0			0
$421$	29.0000			1.41337			0.706687	−	0.707527i	$-0.250189\pi$
$421$	29.0000			1.41337			0.706687	+	0.707527i	$0.250189\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	4.17121	+	2.75699i	0.202333	+	0.133734i
$426$		0			0
$427$	−4.10102	−	4.10102i	−0.198462	−	0.198462i
$428$		0			0
$429$		0			0
$430$		0			0
$431$			33.9732i			1.63643i	0.574910	+	0.818217i	$0.305037\pi$
$431$			33.9732i			1.63643i	−0.574910	+	0.818217i	$0.694963\pi$
$432$		0			0
$433$	−4.34847	+	4.34847i	−0.208974	+	0.208974i	−0.803831	−	0.594857i	$-0.797209\pi$
$433$	−4.34847	+	4.34847i	−0.208974	+	0.208974i	0.594857	+	0.803831i	$0.297209\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	5.97469	−	5.97469i	0.285808	−	0.285808i
$438$		0			0
$439$			17.5505i			0.837640i	0.908069	+	0.418820i	$0.137556\pi$
$439$			17.5505i			0.837640i	−0.908069	+	0.418820i	$0.862444\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−3.46410	−	3.46410i	−0.164584	−	0.164584i	0.620010	−	0.784594i	$-0.287129\pi$
$443$	−3.46410	−	3.46410i	−0.164584	−	0.164584i	−0.784594	+	0.620010i	$0.787129\pi$
$444$		0			0
$445$	−0.202041	+	0.247449i	−0.00957766	+	0.0117302i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	3.46410			0.163481			0.0817405	−	0.996654i	$-0.473952\pi$
$449$	3.46410			0.163481			0.0817405	+	0.996654i	$0.473952\pi$
$450$		0			0
$451$	−18.7980			−0.885161
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−6.92820	+	8.48528i	−0.324799	+	0.397796i
$456$		0			0
$457$	−3.89898	−	3.89898i	−0.182387	−	0.182387i	0.610008	−	0.792395i	$-0.291166\pi$
$457$	−3.89898	−	3.89898i	−0.182387	−	0.182387i	−0.792395	+	0.610008i	$0.791166\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$			22.8345i			1.06351i	0.846899	+	0.531754i	$0.178467\pi$
$461$			22.8345i			1.06351i	−0.846899	+	0.531754i	$0.821533\pi$
$462$		0			0
$463$	−1.10102	+	1.10102i	−0.0511688	+	0.0511688i	−0.732228	−	0.681059i	$-0.761520\pi$
$463$	−1.10102	+	1.10102i	−0.0511688	+	0.0511688i	0.681059	+	0.732228i	$0.261520\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−8.58882	+	8.58882i	−0.397443	+	0.397443i	−0.877330	−	0.479887i	$-0.840678\pi$
$467$	−8.58882	+	8.58882i	−0.397443	+	0.397443i	0.479887	+	0.877330i	$0.340678\pi$
$468$		0			0
$469$		−	10.8990i		−	0.503268i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−7.70674	−	7.70674i	−0.354356	−	0.354356i
$474$		0			0
$475$	4.89898	−	1.00000i	0.224781	−	0.0458831i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−6.00680			−0.274458			−0.137229	−	0.990539i	$-0.543820\pi$
$479$	−6.00680			−0.274458			−0.137229	+	0.990539i	$0.543820\pi$
$480$		0			0
$481$	12.8990			0.588143
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−15.4135	+	1.55708i	−0.699890	+	0.0707033i
$486$		0			0
$487$	−25.7980	−	25.7980i	−1.16902	−	1.16902i	−0.982440	−	0.186577i	$-0.940261\pi$
$487$	−25.7980	−	25.7980i	−1.16902	−	1.16902i	−0.186577	−	0.982440i	$-0.559739\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		−	26.0915i		−	1.17749i	−0.808317	−	0.588747i	$-0.799621\pi$
$491$		−	26.0915i		−	1.17749i	0.808317	−	0.588747i	$-0.200379\pi$
$492$		0			0
$493$	−4.22474	+	4.22474i	−0.190273	+	0.190273i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−0.778539	+	0.778539i	−0.0349223	+	0.0349223i
$498$		0			0
$499$		−	23.5959i		−	1.05630i	−0.849152	−	0.528149i	$-0.822886\pi$
$499$		−	23.5959i		−	1.05630i	0.849152	−	0.528149i	$-0.177114\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	12.2672	+	12.2672i	0.546968	+	0.546968i	0.925563	−	0.378594i	$-0.123592\pi$
$503$	12.2672	+	12.2672i	0.546968	+	0.546968i	−0.378594	+	0.925563i	$0.623592\pi$
$504$		0			0
$505$	−2.69694	−	2.20204i	−0.120012	−	0.0979895i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−40.9335			−1.81435			−0.907174	−	0.420756i	$-0.861765\pi$
$509$	−40.9335			−1.81435			−0.907174	+	0.420756i	$0.861765\pi$
$510$		0			0
$511$	−5.69694			−0.252018
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−1.73205	−	17.1455i	−0.0763233	−	0.755523i
$516$		0			0
$517$	−14.1237	−	14.1237i	−0.621161	−	0.621161i
$518$		0			0
$519$		0			0
$520$		0			0
$521$			15.0956i			0.661352i	0.943744	+	0.330676i	$0.107277\pi$
$521$			15.0956i			0.661352i	−0.943744	+	0.330676i	$0.892723\pi$
$522$		0			0
$523$	8.44949	−	8.44949i	0.369470	−	0.369470i	−0.497814	−	0.867284i	$-0.665864\pi$
$523$	8.44949	−	8.44949i	0.369470	−	0.369470i	0.867284	+	0.497814i	$0.165864\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−3.78194	+	3.78194i	−0.164744	+	0.164744i
$528$		0			0
$529$			48.3939i			2.10408i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	11.9494	+	11.9494i	0.517585	+	0.517585i
$534$		0			0
$535$	−2.10102	−	20.7980i	−0.0908350	−	0.899174i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−12.5851			−0.542077
$540$		0			0
$541$	20.4495			0.879192			0.439596	−	0.898196i	$-0.355122\pi$
$541$	20.4495			0.879192			0.439596	+	0.898196i	$0.355122\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	5.37113	+	4.38551i	0.230074	+	0.187854i
$546$		0			0
$547$	−26.1237	−	26.1237i	−1.11697	−	1.11697i	−0.992184	−	0.124786i	$-0.960176\pi$
$547$	−26.1237	−	26.1237i	−1.11697	−	1.11697i	−0.124786	−	0.992184i	$-0.539824\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$			5.97469i			0.254530i
$552$		0			0
$553$	−9.24745	+	9.24745i	−0.393242	+	0.393242i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	4.24264	−	4.24264i	0.179766	−	0.179766i	−0.611488	−	0.791254i	$-0.709429\pi$
$557$	4.24264	−	4.24264i	0.179766	−	0.179766i	0.791254	+	0.611488i	$0.209429\pi$
$558$		0			0
$559$			9.79796i			0.414410i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	15.5170	+	15.5170i	0.653965	+	0.653965i	0.953945	−	0.299981i	$-0.0969802\pi$
$563$	15.5170	+	15.5170i	0.653965	+	0.653965i	−0.299981	+	0.953945i	$0.596980\pi$
$564$		0			0
$565$	−20.7980	+	2.10102i	−0.874977	+	0.0883906i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−24.5344			−1.02854			−0.514269	−	0.857629i	$-0.671937\pi$
$569$	−24.5344			−1.02854			−0.514269	+	0.857629i	$0.671937\pi$
$570$		0			0
$571$	42.6969			1.78681			0.893406	−	0.449251i	$-0.148309\pi$
$571$	42.6969			1.78681			0.893406	+	0.449251i	$0.148309\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−23.2952	+	35.2446i	−0.971477	+	1.46980i
$576$		0			0
$577$	1.89898	+	1.89898i	0.0790556	+	0.0790556i	0.745529	−	0.666473i	$-0.232197\pi$
$577$	1.89898	+	1.89898i	0.0790556	+	0.0790556i	−0.666473	+	0.745529i	$0.732197\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$			16.9706i			0.704058i
$582$		0			0
$583$	−7.67423	+	7.67423i	−0.317834	+	0.317834i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−11.5601	+	11.5601i	−0.477137	+	0.477137i	−0.904215	−	0.427078i	$-0.859543\pi$
$587$	−11.5601	+	11.5601i	−0.477137	+	0.477137i	0.427078	+	0.904215i	$0.359543\pi$
$588$		0			0
$589$			5.34847i			0.220380i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	32.7733	+	32.7733i	1.34584	+	1.34584i	0.890128	+	0.455711i	$0.150615\pi$
$593$	32.7733	+	32.7733i	1.34584	+	1.34584i	0.455711	+	0.890128i	$0.349385\pi$
$594$		0			0
$595$	−2.44949	+	3.00000i	−0.100419	+	0.122988i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−6.78534			−0.277241			−0.138621	−	0.990346i	$-0.544267\pi$
$599$	−6.78534			−0.277241			−0.138621	+	0.990346i	$0.544267\pi$
$600$		0			0
$601$	−2.24745			−0.0916753			−0.0458377	−	0.998949i	$-0.514596\pi$
$601$	−2.24745			−0.0916753			−0.0458377	+	0.998949i	$0.514596\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	1.55708	−	1.90702i	0.0633042	−	0.0775315i
$606$		0			0
$607$	−3.87628	−	3.87628i	−0.157333	−	0.157333i	0.624051	−	0.781384i	$-0.285486\pi$
$607$	−3.87628	−	3.87628i	−0.157333	−	0.157333i	−0.781384	+	0.624051i	$0.785486\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$			17.9562i			0.726429i
$612$		0			0
$613$	−13.1464	+	13.1464i	−0.530979	+	0.530979i	−0.920864	−	0.389885i	$-0.872515\pi$
$613$	−13.1464	+	13.1464i	−0.530979	+	0.530979i	0.389885	+	0.920864i	$0.372515\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−19.4812	+	19.4812i	−0.784282	+	0.784282i	−0.980550	−	0.196269i	$-0.937118\pi$
$617$	−19.4812	+	19.4812i	−0.784282	+	0.784282i	0.196269	+	0.980550i	$0.437118\pi$
$618$		0			0
$619$			40.9444i			1.64569i	0.568263	+	0.822847i	$0.307616\pi$
$619$			40.9444i			1.64569i	−0.568263	+	0.822847i	$0.692384\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−0.174973	−	0.174973i	−0.00701013	−	0.00701013i
$624$		0			0
$625$	−23.0000	+	9.79796i	−0.920000	+	0.391918i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	4.56048			0.181838
$630$		0			0
$631$	43.8990			1.74759			0.873795	−	0.486294i	$-0.161652\pi$
$631$	43.8990			1.74759			0.873795	+	0.486294i	$0.161652\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−12.9029	+	1.30346i	−0.512036	+	0.0517261i
$636$		0			0
$637$	8.00000	+	8.00000i	0.316972	+	0.316972i
$638$		0			0
$639$		0			0
$640$		0			0
$641$			12.6172i			0.498348i	0.968459	+	0.249174i	$0.0801591\pi$
$641$			12.6172i			0.498348i	−0.968459	+	0.249174i	$0.919841\pi$
$642$		0			0
$643$	−9.22474	+	9.22474i	−0.363788	+	0.363788i	−0.865206	−	0.501417i	$-0.832812\pi$
$643$	−9.22474	+	9.22474i	−0.363788	+	0.363788i	0.501417	+	0.865206i	$0.332812\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	1.55708	−	1.55708i	0.0612151	−	0.0612151i	−0.675837	−	0.737052i	$-0.736217\pi$
$647$	1.55708	−	1.55708i	0.0612151	−	0.0612151i	0.737052	+	0.675837i	$0.236217\pi$
$648$		0			0
$649$		−	17.3485i		−	0.680987i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−10.3923	−	10.3923i	−0.406682	−	0.406682i	0.473898	−	0.880580i	$-0.342847\pi$
$653$	−10.3923	−	10.3923i	−0.406682	−	0.406682i	−0.880580	+	0.473898i	$0.842847\pi$
$654$		0			0
$655$	13.1010	+	10.6969i	0.511899	+	0.417964i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	19.0205			0.740932			0.370466	−	0.928846i	$-0.379198\pi$
$659$	19.0205			0.740932			0.370466	+	0.928846i	$0.379198\pi$
$660$		0			0
$661$	14.3031			0.556325			0.278162	−	0.960534i	$-0.410275\pi$
$661$	14.3031			0.556325			0.278162	+	0.960534i	$0.410275\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	0.389270	+	3.85337i	0.0150952	+	0.149427i
$666$		0			0
$667$	−35.6969	−	35.6969i	−1.38219	−	1.38219i
$668$		0			0
$669$		0			0
$670$		0			0
$671$			10.5352i			0.406706i
$672$		0			0
$673$	−30.0454	+	30.0454i	−1.15817	+	1.15817i	−0.173295	+	0.984870i	$0.555442\pi$
$673$	−30.0454	+	30.0454i	−1.15817	+	1.15817i	−0.984870	+	0.173295i	$0.944558\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−18.7347	+	18.7347i	−0.720034	+	0.720034i	−0.968612	−	0.248578i	$-0.920037\pi$
$677$	−18.7347	+	18.7347i	−0.720034	+	0.720034i	0.248578	+	0.968612i	$0.420037\pi$
$678$		0			0
$679$		−	12.0000i		−	0.460518i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−7.70674	−	7.70674i	−0.294890	−	0.294890i	0.544118	−	0.839009i	$-0.316864\pi$
$683$	−7.70674	−	7.70674i	−0.294890	−	0.294890i	−0.839009	+	0.544118i	$0.816864\pi$
$684$		0			0
$685$	−2.07832	−	20.5732i	−0.0794084	−	0.786062i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	9.75663			0.371698
$690$		0			0
$691$	7.34847			0.279549			0.139774	−	0.990183i	$-0.455362\pi$
$691$	7.34847			0.279549			0.139774	+	0.990183i	$0.455362\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−4.24264	−	3.46410i	−0.160933	−	0.131401i
$696$		0			0
$697$	4.22474	+	4.22474i	0.160024	+	0.160024i
$698$		0			0
$699$		0			0
$700$		0			0
$701$		−	22.8345i		−	0.862447i	−0.902245	−	0.431224i	$-0.858082\pi$
$701$		−	22.8345i		−	0.862447i	0.902245	−	0.431224i	$-0.141918\pi$
$702$		0			0
$703$	3.22474	−	3.22474i	0.121624	−	0.121624i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	1.90702	−	1.90702i	0.0717210	−	0.0717210i
$708$		0			0
$709$			51.1464i			1.92084i	0.278548	+	0.960422i	$0.410147\pi$
$709$			51.1464i			1.92084i	−0.278548	+	0.960422i	$0.589853\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−31.9555	−	31.9555i	−1.19674	−	1.19674i
$714$		0			0
$715$	19.7980	−	2.00000i	0.740402	−	0.0747958i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−29.3020			−1.09278			−0.546390	−	0.837531i	$-0.683998\pi$
$719$	−29.3020			−1.09278			−0.546390	+	0.837531i	$0.683998\pi$
$720$		0			0
$721$	13.3485			0.497123
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−5.97469	−	29.2699i	−0.221894	−	1.08706i
$726$		0			0
$727$	12.3485	+	12.3485i	0.457979	+	0.457979i	0.897992	−	0.440012i	$-0.145026\pi$
$727$	12.3485	+	12.3485i	0.457979	+	0.457979i	−0.440012	+	0.897992i	$0.645026\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$			3.46410i			0.128124i
$732$		0			0
$733$	−19.9444	+	19.9444i	−0.736663	+	0.736663i	−0.971931	−	0.235268i	$-0.924403\pi$
$733$	−19.9444	+	19.9444i	−0.736663	+	0.736663i	0.235268	+	0.971931i	$0.424403\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−13.9993	+	13.9993i	−0.515670	+	0.515670i
$738$		0			0
$739$			35.7980i			1.31685i	0.752647	+	0.658425i	$0.228777\pi$
$739$			35.7980i			1.31685i	−0.752647	+	0.658425i	$0.771223\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	11.7744	+	11.7744i	0.431961	+	0.431961i	0.889295	−	0.457334i	$-0.151196\pi$
$743$	11.7744	+	11.7744i	0.431961	+	0.431961i	−0.457334	+	0.889295i	$0.651196\pi$
$744$		0			0
$745$	−2.00000	+	2.44949i	−0.0732743	+	0.0897424i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	16.1920			0.591644
$750$		0			0
$751$	51.1918			1.86802			0.934008	−	0.357251i	$-0.116286\pi$
$751$	51.1918			1.86802			0.934008	+	0.357251i	$0.116286\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−2.68556	+	3.28913i	−0.0977376	+	0.119704i
$756$		0			0
$757$	3.00000	+	3.00000i	0.109037	+	0.109037i	0.759520	−	0.650484i	$-0.225434\pi$
$757$	3.00000	+	3.00000i	0.109037	+	0.109037i	−0.650484	+	0.759520i	$0.725434\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		−	15.1278i		−	0.548381i	−0.961675	−	0.274190i	$-0.911590\pi$
$761$		−	15.1278i		−	0.548381i	0.961675	−	0.274190i	$-0.0884098\pi$
$762$		0			0
$763$	−3.79796	+	3.79796i	−0.137495	+	0.137495i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−11.0280	+	11.0280i	−0.398197	+	0.398197i
$768$		0			0
$769$		−	38.5959i		−	1.39180i	−0.718137	−	0.695902i	$-0.755005\pi$
$769$		−	38.5959i		−	1.39180i	0.718137	−	0.695902i	$-0.244995\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	6.82466	+	6.82466i	0.245466	+	0.245466i	0.819107	−	0.573641i	$-0.194469\pi$
$773$	6.82466	+	6.82466i	0.245466	+	0.245466i	−0.573641	+	0.819107i	$0.694469\pi$
$774$		0			0
$775$	−5.34847	−	26.2020i	−0.192123	−	0.941205i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	5.97469			0.214066
$780$		0			0
$781$	2.00000			0.0715656
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−37.4373	+	3.78194i	−1.33620	+	0.134983i
$786$		0			0
$787$	16.3712	+	16.3712i	0.583569	+	0.583569i	0.935882	−	0.352313i	$-0.114605\pi$
$787$	16.3712	+	16.3712i	0.583569	+	0.583569i	−0.352313	+	0.935882i	$0.614605\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		−	16.1920i		−	0.575722i
$792$		0			0
$793$	6.69694	−	6.69694i	0.237815	−	0.237815i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	15.9456	−	15.9456i	0.564823	−	0.564823i	−0.365851	−	0.930674i	$-0.619222\pi$
$797$	15.9456	−	15.9456i	0.564823	−	0.564823i	0.930674	+	0.365851i	$0.119222\pi$
$798$		0			0
$799$			6.34847i			0.224593i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	7.31747	+	7.31747i	0.258228	+	0.258228i
$804$		0			0
$805$	−25.3485	−	20.6969i	−0.893416	−	0.729471i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−22.3417			−0.785492			−0.392746	−	0.919647i	$-0.628475\pi$
$809$	−22.3417			−0.785492			−0.392746	+	0.919647i	$0.628475\pi$
$810$		0			0
$811$	20.2474			0.710984			0.355492	−	0.934679i	$-0.384313\pi$
$811$	20.2474			0.710984			0.355492	+	0.934679i	$0.384313\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	1.37489	+	13.6100i	0.0481603	+	0.476738i
$816$		0			0
$817$	2.44949	+	2.44949i	0.0856968	+	0.0856968i
$818$		0			0
$819$		0			0
$820$		0			0
$821$			0.921404i			0.0321572i	0.999871	+	0.0160786i	$0.00511820\pi$
$821$			0.921404i			0.0321572i	−0.999871	+	0.0160786i	$0.994882\pi$
$822$		0			0
$823$	24.6186	−	24.6186i	0.858151	−	0.858151i	−0.132969	−	0.991120i	$-0.542451\pi$
$823$	24.6186	−	24.6186i	0.858151	−	0.858151i	0.991120	+	0.132969i	$0.0424511\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−6.89610	+	6.89610i	−0.239801	+	0.239801i	−0.816768	−	0.576967i	$-0.804236\pi$
$827$	−6.89610	+	6.89610i	−0.239801	+	0.239801i	0.576967	+	0.816768i	$0.304236\pi$
$828$		0			0
$829$			12.5959i			0.437474i	0.975784	+	0.218737i	$0.0701937\pi$
$829$			12.5959i			0.437474i	−0.975784	+	0.218737i	$0.929806\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	2.82843	+	2.82843i	0.0979992	+	0.0979992i
$834$		0			0
$835$	1.50510	+	14.8990i	0.0520862	+	0.515600i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	17.9241			0.618808			0.309404	−	0.950931i	$-0.399871\pi$
$839$	17.9241			0.618808			0.309404	+	0.950931i	$0.399871\pi$
$840$		0			0
$841$	6.69694			0.230929
$842$		0			0
$843$		0			0
$844$		0			0
$845$	8.66025	+	7.07107i	0.297922	+	0.243252i
$846$		0			0
$847$	1.34847	+	1.34847i	0.0463340	+	0.0463340i
$848$		0			0
$849$		0			0
$850$		0			0
$851$			38.5337i			1.32092i
$852$		0			0
$853$	10.0454	−	10.0454i	0.343948	−	0.343948i	−0.513901	−	0.857849i	$-0.671800\pi$
$853$	10.0454	−	10.0454i	0.343948	−	0.343948i	0.857849	+	0.513901i	$0.171800\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−12.7279	+	12.7279i	−0.434778	+	0.434778i	−0.890250	−	0.455472i	$-0.849470\pi$
$857$	−12.7279	+	12.7279i	−0.434778	+	0.434778i	0.455472	+	0.890250i	$0.349470\pi$
$858$		0			0
$859$			21.2020i			0.723404i	0.932294	+	0.361702i	$0.117804\pi$
$859$			21.2020i			0.723404i	−0.932294	+	0.361702i	$0.882196\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−26.1951	−	26.1951i	−0.891690	−	0.891690i	0.102992	−	0.994682i	$-0.467158\pi$
$863$	−26.1951	−	26.1951i	−0.891690	−	0.891690i	−0.994682	+	0.102992i	$0.967158\pi$
$864$		0			0
$865$	15.3485	−	1.55051i	0.521864	−	0.0527189i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	23.7559			0.805864
$870$		0			0
$871$	17.7980			0.603061
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−5.76039	−	18.4883i	−0.194737	−	0.625019i
$876$		0			0
$877$	22.5732	+	22.5732i	0.762243	+	0.762243i	0.976727	−	0.214484i	$-0.0688070\pi$
$877$	22.5732	+	22.5732i	0.762243	+	0.762243i	−0.214484	+	0.976727i	$0.568807\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		−	7.81750i		−	0.263378i	−0.991291	−	0.131689i	$-0.957960\pi$
$881$		−	7.81750i		−	0.263378i	0.991291	−	0.131689i	$-0.0420401\pi$
$882$		0			0
$883$	21.5959	−	21.5959i	0.726761	−	0.726761i	−0.243213	−	0.969973i	$-0.578201\pi$
$883$	21.5959	−	21.5959i	0.726761	−	0.726761i	0.969973	+	0.243213i	$0.0782012\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−38.8515	+	38.8515i	−1.30451	+	1.30451i	−0.379187	+	0.925320i	$0.623796\pi$
$887$	−38.8515	+	38.8515i	−1.30451	+	1.30451i	−0.925320	+	0.379187i	$0.876204\pi$
$888$		0			0
$889$		−	10.0454i		−	0.336912i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	4.48905	+	4.48905i	0.150220	+	0.150220i
$894$		0			0
$895$	31.5959	−	38.6969i	1.05614	−	1.29350i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	31.9555			1.06577
$900$		0			0
$901$	3.44949			0.114919
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−9.75663	+	11.9494i	−0.324321	+	0.397211i
$906$		0			0
$907$	0.977296	+	0.977296i	0.0324506	+	0.0324506i	0.723146	−	0.690695i	$-0.242695\pi$
$907$	0.977296	+	0.977296i	0.0324506	+	0.0324506i	−0.690695	+	0.723146i	$0.742695\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$			10.4244i			0.345376i	0.984977	+	0.172688i	$0.0552453\pi$
$911$			10.4244i			0.345376i	−0.984977	+	0.172688i	$0.944755\pi$
$912$		0			0
$913$	21.7980	−	21.7980i	0.721407	−	0.721407i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−9.26382	+	9.26382i	−0.305918	+	0.305918i
$918$		0			0
$919$			21.8990i			0.722381i	0.932492	+	0.361190i	$0.117630\pi$
$919$			21.8990i			0.722381i	−0.932492	+	0.361190i	$0.882370\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−1.27135	−	1.27135i	−0.0418470	−	0.0418470i
$924$		0			0
$925$	−12.5732	+	19.0227i	−0.413405	+	0.625463i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	32.7019			1.07291			0.536457	−	0.843928i	$-0.319762\pi$
$929$	32.7019			1.07291			0.536457	+	0.843928i	$0.319762\pi$
$930$		0			0
$931$	4.00000			0.131095
$932$		0			0
$933$		0			0
$934$		0			0
$935$	6.99964	−	0.707107i	0.228913	−	0.0231249i
$936$		0			0
$937$	28.4495	+	28.4495i	0.929404	+	0.929404i	0.997667	−	0.0682630i	$-0.0217457\pi$
$937$	28.4495	+	28.4495i	0.929404	+	0.929404i	−0.0682630	+	0.997667i	$0.521746\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		−	38.7408i		−	1.26291i	−0.775411	−	0.631457i	$-0.782457\pi$
$941$		−	38.7408i		−	1.26291i	0.775411	−	0.631457i	$-0.217543\pi$
$942$		0			0
$943$	−35.6969	+	35.6969i	−1.16245	+	1.16245i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	30.3663	−	30.3663i	0.986771	−	0.986771i	−0.0131425	−	0.999914i	$-0.504184\pi$
$947$	30.3663	−	30.3663i	0.986771	−	0.986771i	0.999914	+	0.0131425i	$0.00418352\pi$
$948$		0			0
$949$		−	9.30306i		−	0.301990i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	0.389270	+	0.389270i	0.0126097	+	0.0126097i	0.713384	−	0.700774i	$-0.247162\pi$
$953$	0.389270	+	0.389270i	0.0126097	+	0.0126097i	−0.700774	+	0.713384i	$0.747162\pi$
$954$		0			0
$955$	−20.4495	−	16.6969i	−0.661730	−	0.540300i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	16.0171			0.517218
$960$		0			0
$961$	−2.39388			−0.0772218
$962$		0			0
$963$		0			0
$964$		0			0
$965$	1.12848	+	11.1708i	0.0363272	+	0.359602i
$966$		0			0
$967$	21.3939	+	21.3939i	0.687981	+	0.687981i	0.961785	−	0.273805i	$-0.0882822\pi$
$967$	21.3939	+	21.3939i	0.687981	+	0.687981i	−0.273805	+	0.961785i	$0.588282\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		−	17.8920i		−	0.574181i	−0.957904	−	0.287090i	$-0.907312\pi$
$971$		−	17.8920i		−	0.574181i	0.957904	−	0.287090i	$-0.0926880\pi$
$972$		0			0
$973$	3.00000	−	3.00000i	0.0961756	−	0.0961756i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	25.0273	−	25.0273i	0.800693	−	0.800693i	−0.182511	−	0.983204i	$-0.558423\pi$
$977$	25.0273	−	25.0273i	0.800693	−	0.800693i	0.983204	+	0.182511i	$0.0584226\pi$
$978$		0			0
$979$			0.449490i			0.0143658i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	35.6732	+	35.6732i	1.13780	+	1.13780i	0.988844	+	0.148953i	$0.0475904\pi$
$983$	35.6732	+	35.6732i	1.13780	+	1.13780i	0.148953	+	0.988844i	$0.452410\pi$
$984$		0			0
$985$	−6.10102	−	60.3939i	−0.194395	−	1.92431i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−29.2699			−0.930728
$990$		0			0
$991$	−30.8990			−0.981538			−0.490769	−	0.871290i	$-0.663284\pi$
$991$	−30.8990			−0.981538			−0.490769	+	0.871290i	$0.663284\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	42.7764	+	34.9267i	1.35610	+	1.10725i
$996$		0			0
$997$	−1.47219	−	1.47219i	−0.0466248	−	0.0466248i	0.683410	−	0.730035i	$-0.260496\pi$
$997$	−1.47219	−	1.47219i	−0.0466248	−	0.0466248i	−0.730035	+	0.683410i	$0.760496\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3060.2.w.d.2177.3	yes	8
3.2	odd	2	inner	3060.2.w.d.2177.2	yes	8
5.3	odd	4	inner	3060.2.w.d.953.1	✓	8
15.8	even	4	inner	3060.2.w.d.953.4	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3060.2.w.d.953.1	✓	8	5.3	odd	4	inner
3060.2.w.d.953.4	yes	8	15.8	even	4	inner
3060.2.w.d.2177.2	yes	8	3.2	odd	2	inner
3060.2.w.d.2177.3	yes	8	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 3060 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3060.w (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(1.41421 - 1.73205i) q^{5} +(1.22474 + 1.22474i) q^{7} -3.14626i q^{11} +(-2.00000 + 2.00000i) q^{13} +(-0.707107 + 0.707107i) q^{17} +1.00000i q^{19} +(-5.97469 - 5.97469i) q^{23} +(-1.00000 - 4.89898i) q^{25} +5.97469 q^{29} +5.34847 q^{31} +(3.85337 - 0.389270i) q^{35} +(-3.22474 - 3.22474i) q^{37} -5.97469i q^{41} +(2.44949 - 2.44949i) q^{43} +(4.48905 - 4.48905i) q^{47} -4.00000i q^{49} +(-2.43916 - 2.43916i) q^{53} +(-5.44949 - 4.44949i) q^{55} +5.51399 q^{59} -3.34847 q^{61} +(0.635674 + 6.29253i) q^{65} +(-4.44949 - 4.44949i) q^{67} +0.635674i q^{71} +(-2.32577 + 2.32577i) q^{73} +(3.85337 - 3.85337i) q^{77} +7.55051i q^{79} +(6.92820 + 6.92820i) q^{83} +(0.224745 + 2.22474i) q^{85} -0.142865 q^{89} -4.89898 q^{91} +(1.73205 + 1.41421i) q^{95} +(-4.89898 - 4.89898i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 16 q^{13} - 8 q^{25} - 16 q^{31} - 16 q^{37} - 24 q^{55} + 32 q^{61} - 16 q^{67} - 48 q^{73} - 8 q^{85}+O(q^{100}) \) 8 * q - 16 * q^13 - 8 * q^25 - 16 * q^31 - 16 * q^37 - 24 * q^55 + 32 * q^61 - 16 * q^67 - 48 * q^73 - 8 * q^85

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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